Displacement-measuring interferometers monitor changes in the position of a measurement object relative to a reference object based on an optical interference signal. The interferometer generates the optical interference signal by overlapping and interfering a measurement beam reflected from the measurement object with a reference beam reflected from the reference object.
In many applications, the measurement and reference beams have orthogonal polarizations and different frequencies. The different frequencies can be produced, for example, by laser Zeeman splitting, by acousto-optical modulation, or internal to the laser using birefringent elements or the like. The orthogonal polarizations allow a polarizing beam splitter to direct the measurement and reference beams to the measurement and reference objects, respectively, and combine the reflected measurement and reference beams to form overlapping exit measurement and reference beams. The overlapping exit beams form an output beam that subsequently passes through a polarizer.
The polarizer mixes polarizations of the exit measurement and reference beams to form a mixed beam. Components of the exit measurement and reference beams in the mixed beam interfere with one another so that the intensity of the mixed beam varies with the relative phase of the exit measurement and reference beams. A detector measures the time-dependent intensity of the mixed beam and generates an electrical interference signal proportional to that intensity. Because the measurement and reference beams have different frequencies, the electrical interference signal includes a “heterodyne” signal having a beat frequency equal to the difference between the frequencies of the exit measurement and reference beams. If the lengths of the measurement and reference paths are changing relative to one another, e.g., by translating a stage that includes the measurement object, the measured beat frequency includes a Doppler shift equal to 2νnp/λ, where ν is the relative speed of the measurement and reference objects, λ is the wavelength of the measurement and reference beams, n is the refractive index of the medium through which the light beams travel, e.g., air or vacuum, and p is the number of passes to the reference and measurement objects. Changes in the relative position of the measurement object correspond to changes in the phase of the measured interference signal, with a 2π phase change substantially equal to a distance change L of λ/(np), where L is a round-trip distance change, e.g., the change in distance to and from a stage that includes the measurement object.
Unfortunately, this equality is not always exact. In addition, the amplitude of the measured interference signal may be variable. A variable amplitude may subsequently reduce the accuracy of measured phase changes. Many interferometers include non-linearities such as what are known as “cyclic errors.” The cyclic errors can be expressed as contributions, to the phase and/or the intensity of the measured interference signal and have a sinusoidal dependence on the change in optical path length pnL. In particular, the first harmonic cyclic error in phase has a sinusoidal dependence on (2πpnL)/λ and the second harmonic cyclic error in phase has a sinusoidal dependence on 2 (2πpnL)/λ. Higher harmonic cyclic errors can also be present.
Another source of errors are related to environmental effects such as air turbulence and non-isotropic distributions of gases in the interferometer evironment. See, for example, an article entitled “Residual Errors In Laser Interferometry From Air Turbulence And Nonlinearity,” by N. Bobroff, Appl. Opt. 26(13), 2676–2682 (1987), and an article entitled “Recent Advances In Displacement Measuring Interferometry,” also by N. Bobroff, Measurement Science & Tech. 4(9), 907–926 (1993). As noted in the aforementioned cited references, interferometric displacement measurements in a gas are subject to environmental uncertainties, particularly to changes in air pressure and temperature; to uncertainties in air composition such as resulting from changes in humidity and/or the presence of additional gases; and to the effects of turbulence in the gas. Such factors alter the wavelength of the light used to measure the displacement. Under normal conditions, the refractive index of air for example is approximately 1.0003 with a variation of the order of 1×10−5 to 1×10−4. In many applications the refractive index of air must be known with a relative precision of less than 0.1 ppm (parts per million) to less than 0.001 ppm, these two relative precisions corresponding to a displacement measurement accuracy of 100 nm and less than 1 nm, respectively, for a one meter interferometric displacement measurement.
One way to detect refractive index fluctuations is to measure changes in pressure and temperature along a measurement path and calculate the effect on the optical path length of the measurement path. Another, more direct way to detect the effects of a fluctuating refractive index over a measurement path is by multiple-wavelength distance measurement. The basic principle may be understood as follows. Interferometers and laser radar measure the optical path length between a reference and an object, most often in open air. The optical path length is the integrated product of the refractive index and the physical path traversed by a measurement beam. In that the refractive index varies with wavelength, but the physical path is independent of wavelength, it is generally possible to determine the physical path length from the optical path length, particularly the contributions of fluctuations in refractive index, provided that the instrument employs at least two wavelengths. The variation of refractive index with wavelength is known in the art as dispersion and this technique is often referred to as the dispersion technique or as dispersion interferometry.